HW3_WhiteWines
Alyssa Monda
2/14/2017
library(readr)
White_wines <- read.table("~/Desktop/Big Data/Regression-1/White_wines.csv", header=TRUE, sep=",", na.strings="NA", dec=".", strip.white=TRUE)
View(White_wines)## Warning: running command ''/usr/bin/otool' -L '/Library/Frameworks/
## R.framework/Resources/modules/R_de.so'' had status 1# Import Data
#View(White_wines)Summary of Data
Look at a summary of the data.
summary(White_wines)## fixed.acidity volatile.acidity citric.acid residual.sugar
## Min. : 3.800 Min. :0.0800 Min. :0.0000 Min. : 0.600
## 1st Qu.: 6.300 1st Qu.:0.2100 1st Qu.:0.2700 1st Qu.: 1.700
## Median : 6.800 Median :0.2600 Median :0.3200 Median : 5.200
## Mean : 6.855 Mean :0.2782 Mean :0.3342 Mean : 6.391
## 3rd Qu.: 7.300 3rd Qu.:0.3200 3rd Qu.:0.3900 3rd Qu.: 9.900
## Max. :14.200 Max. :1.1000 Max. :1.6600 Max. :65.800
## chlorides free.sulfur.dioxide total.sulfur.dioxide
## Min. :0.00900 Min. : 2.00 Min. : 9.0
## 1st Qu.:0.03600 1st Qu.: 23.00 1st Qu.:108.0
## Median :0.04300 Median : 34.00 Median :134.0
## Mean :0.04577 Mean : 35.31 Mean :138.4
## 3rd Qu.:0.05000 3rd Qu.: 46.00 3rd Qu.:167.0
## Max. :0.34600 Max. :289.00 Max. :440.0
## density pH sulphates alcohol
## Min. :0.9871 Min. :2.720 Min. :0.2200 Min. : 8.00
## 1st Qu.:0.9917 1st Qu.:3.090 1st Qu.:0.4100 1st Qu.: 9.50
## Median :0.9937 Median :3.180 Median :0.4700 Median :10.40
## Mean :0.9940 Mean :3.188 Mean :0.4898 Mean :10.51
## 3rd Qu.:0.9961 3rd Qu.:3.280 3rd Qu.:0.5500 3rd Qu.:11.40
## Max. :1.0390 Max. :3.820 Max. :1.0800 Max. :14.20
## quality
## Min. :3.000
## 1st Qu.:5.000
## Median :6.000
## Mean :5.878
## 3rd Qu.:6.000
## Max. :9.000This dataset is composed of 13 variables. The dependent variable of interest is quality. We will investigate the relationship between the remaining variables (fixed acidity, volatile acid, citric acid, residual sugar, chlorides, free sulfur dioxide, total sulfur dioxide, density, pH, sulphates, and alcohol) and quality.
Univariate Analysis
Quality appears to be normally distributed with scores ranging from a minimum of 3 to a maximum of 9, with a mean score of of 5.88 and a median of 6.0. A boxplot of quality shows the potential of outliers. These should be considered when interpretting the remainder of the analysis.
with(White_wines, Hist(quality, scale="frequency", breaks="Sturges",
col="darkgray"))
Boxplot( ~ quality, data=White_wines, id.method="y")
## [1] "252" "254" "295" "446" "741" "874" "1035" "1230" "1418" "1485"
## [11] "775" "821" "828" "877" "1606" "18" "21" "23" "69" "75"The distribiution of the remaining variables can be seen in the histograms below. Residual sugar, alcohol, and volatile acid appear to have right skewed distributions, while not perfect, the other variables appear to have a normal distribution.
with(White_wines, Hist(alcohol, scale="frequency", breaks="Sturges",
col="darkgray"))
with(White_wines, Hist(chlorides, scale="frequency", breaks="Sturges",
col="darkgray"))
with(White_wines, Hist(citric.acid, scale="frequency", breaks="Sturges",
col="darkgray"))
with(White_wines, Hist(density, scale="frequency", breaks="Sturges",
col="darkgray"))
with(White_wines, Hist(fixed.acidity, scale="frequency", breaks="Sturges",
col="darkgray"))
with(White_wines, Hist(free.sulfur.dioxide, scale="frequency",
breaks="Sturges", col="darkgray"))
with(White_wines, Hist(pH, scale="frequency", breaks="Sturges",
col="darkgray"))
with(White_wines, Hist(residual.sugar, scale="frequency", breaks="Sturges",
col="darkgray"))
with(White_wines, Hist(sulphates, scale="frequency", breaks="Sturges",
col="darkgray"))
with(White_wines, Hist(total.sulfur.dioxide, scale="frequency",
breaks="Sturges", col="darkgray"))
with(White_wines, Hist(volatile.acidity, scale="frequency",
breaks="Sturges", col="darkgray"))
Multivariate Analysis
To begin investigating potential relationships scattlot matrices have been run below.
Scatterplot Matrix: quality, alcohol, chlorides, citric acid.
scatterplotMatrix(~alcohol+chlorides+citric.acid+quality, reg.line=FALSE,
smooth=FALSE, spread=FALSE, span=0.5, ellipse=FALSE, levels=c(.5, .9),
id.n=0, diagonal = 'density', data=White_wines)
Scatterplot Matrix: quality, density, fixed acidity, free sulfur dioxide.
scatterplotMatrix(~density+fixed.acidity+free.sulfur.dioxide+quality,
reg.line=FALSE, smooth=FALSE, spread=FALSE, span=0.5, ellipse=FALSE,
levels=c(.5, .9), id.n=0, diagonal = 'density', data=White_wines)
Scatterplot Matrix: quality, pH, residual sugar, sulphates.
scatterplotMatrix(~pH+quality+residual.sugar+sulphates, reg.line=FALSE,
smooth=FALSE, spread=FALSE, span=0.5, ellipse=FALSE, levels=c(.5, .9),
id.n=0, diagonal = 'density', data=White_wines)
Scatterplot Matrix: quality, total sulfur dioxide, volatile acidity.
scatterplotMatrix(~quality+total.sulfur.dioxide+volatile.acidity,
reg.line=FALSE, smooth=FALSE, spread=FALSE, span=0.5, ellipse=FALSE,
levels=c(.5, .9), id.n=0, diagonal = 'density', data=White_wines)
Linear Correlation analysis shows:
cor(White_wines[,c("alcohol","chlorides","citric.acid","density",
"fixed.acidity","quality")], use="complete")## alcohol chlorides citric.acid density
## alcohol 1.00000000 -0.36018871 -0.075728730 -0.7801376
## chlorides -0.36018871 1.00000000 0.114364448 0.2572113
## citric.acid -0.07572873 0.11436445 1.000000000 0.1495026
## density -0.78013762 0.25721132 0.149502571 1.0000000
## fixed.acidity -0.12088112 0.02308564 0.289180698 0.2653310
## quality 0.43557472 -0.20993441 -0.009209091 -0.3071233
## fixed.acidity quality
## alcohol -0.12088112 0.435574715
## chlorides 0.02308564 -0.209934411
## citric.acid 0.28918070 -0.009209091
## density 0.26533101 -0.307123313
## fixed.acidity 1.00000000 -0.113662831
## quality -0.11366283 1.000000000cor(White_wines[,c("free.sulfur.dioxide","pH","quality","residual.sugar",
"sulphates")], use="complete")## free.sulfur.dioxide pH quality
## free.sulfur.dioxide 1.0000000000 -0.0006177961 0.008158067
## pH -0.0006177961 1.0000000000 0.099427246
## quality 0.0081580671 0.0994272457 1.000000000
## residual.sugar 0.2990983537 -0.1941334540 -0.097576829
## sulphates 0.0592172458 0.1559514973 0.053677877
## residual.sugar sulphates
## free.sulfur.dioxide 0.29909835 0.05921725
## pH -0.19413345 0.15595150
## quality -0.09757683 0.05367788
## residual.sugar 1.00000000 -0.02666437
## sulphates -0.02666437 1.00000000cor(White_wines[,c("quality","total.sulfur.dioxide","volatile.acidity")],
use="complete")## quality total.sulfur.dioxide volatile.acidity
## quality 1.0000000 -0.1747372 -0.1947230
## total.sulfur.dioxide -0.1747372 1.0000000 0.0892605
## volatile.acidity -0.1947230 0.0892605 1.0000000Variable Correlation With quality (r)
alcohol 0.435574715 chlorides -0.209934411 citric.acid -0.009209091 density -0.307123313 fixed.acidity -0.113662831 free.sulfur.dioxide 0.008158067 pH 0.099427246 residual.sugar -0.097576829 sulphates 0.053677877 total.sulfur.dioxide -0.1747372 volatile.acidity -0.1947230
There seems to be a weak positive relationship between alcohol and quality. Density, chlorides, total sulfur dioxide, and volatile acid, seem to have the strongest negative correlations with quality.
Linear Regressions
To further investigate potential relationships between quality and the variables linear regressions have been run below.
Regression Model Alcohol and quality.
RegModel.Alcohol <- lm(alcohol~quality, data=White_wines)
summary(RegModel.Alcohol)##
## Call:
## lm(formula = alcohol ~ quality, data = White_wines)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.2986 -0.7882 -0.1382 0.8014 4.1223
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.95670 0.10626 65.47 <2e-16 ***
## quality 0.60524 0.01788 33.86 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.108 on 4896 degrees of freedom
## Multiple R-squared: 0.1897, Adjusted R-squared: 0.1896
## F-statistic: 1146 on 1 and 4896 DF, p-value: < 2.2e-16Regression Model fixed.acidity and quality.
RegModel.fixed.acidity <- lm(fixed.acidity~quality, data=White_wines)
summary(RegModel.fixed.acidity)##
## Call:
## lm(formula = fixed.acidity ~ quality, data = White_wines)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.0416 -0.5499 -0.0499 0.4667 7.3584
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.49138 0.08042 93.152 < 2e-16 ***
## quality -0.10830 0.01353 -8.005 1.48e-15 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8385 on 4896 degrees of freedom
## Multiple R-squared: 0.01292, Adjusted R-squared: 0.01272
## F-statistic: 64.08 on 1 and 4896 DF, p-value: 1.48e-15Regression Model volatile.acidity and quality.
RegModel.volatile.acidity <- lm(volatile.acidity~quality, data=White_wines)
summary(RegModel.volatile.acidity)##
## Call:
## lm(formula = volatile.acidity ~ quality, data = White_wines)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.20986 -0.06554 -0.01554 0.04446 0.78014
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.408504 0.009483 43.08 <2e-16 ***
## quality -0.022161 0.001595 -13.89 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.09888 on 4896 degrees of freedom
## Multiple R-squared: 0.03792, Adjusted R-squared: 0.03772
## F-statistic: 193 on 1 and 4896 DF, p-value: < 2.2e-16Regression Model citric.acid and quality.
RegModel.citric.acid <- lm(citric.acid~quality, data=White_wines)
summary(RegModel.citric.acid)##
## Call:
## lm(formula = citric.acid ~ quality, data = White_wines)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.3366 -0.0653 -0.0153 0.0547 1.3260
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.341588 0.011608 29.427 <2e-16 ***
## quality -0.001258 0.001953 -0.644 0.519
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.121 on 4896 degrees of freedom
## Multiple R-squared: 8.481e-05, Adjusted R-squared: -0.0001194
## F-statistic: 0.4153 on 1 and 4896 DF, p-value: 0.5193Regression Model residual.sugar and quality.
RegModel.residual.sugar <- lm(residual.sugar~quality, data=White_wines)
summary(RegModel.residual.sugar)##
## Call:
## lm(formula = residual.sugar ~ quality, data = White_wines)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.300 -4.482 -1.023 3.412 59.477
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.67613 0.48420 19.98 < 2e-16 ***
## quality -0.55882 0.08146 -6.86 7.72e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.048 on 4896 degrees of freedom
## Multiple R-squared: 0.009521, Adjusted R-squared: 0.009319
## F-statistic: 47.06 on 1 and 4896 DF, p-value: 7.724e-12Regression Model chlorides and quality.
RegModel.chlorides <- lm(chlorides~quality, data=White_wines)
summary(RegModel.chlorides)##
## Call:
## lm(formula = chlorides ~ quality, data = White_wines)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.042498 -0.009319 -0.003140 0.003860 0.295681
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0762135 0.0020490 37.20 <2e-16 ***
## quality -0.0051789 0.0003447 -15.02 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.02136 on 4896 degrees of freedom
## Multiple R-squared: 0.04407, Adjusted R-squared: 0.04388
## F-statistic: 225.7 on 1 and 4896 DF, p-value: < 2.2e-16Regression Model free.sulfur.dioxide and quality.
RegModel.free.sulfur.dioxide <- lm(free.sulfur.dioxide~quality, data=White_wines)
summary(RegModel.free.sulfur.dioxide)##
## Call:
## lm(formula = free.sulfur.dioxide ~ quality, data = White_wines)
##
## Residuals:
## Min 1Q Median 3Q Max
## -33.171 -12.171 -1.484 10.516 254.143
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 34.3872 1.6313 21.080 <2e-16 ***
## quality 0.1567 0.2744 0.571 0.568
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 17.01 on 4896 degrees of freedom
## Multiple R-squared: 6.655e-05, Adjusted R-squared: -0.0001377
## F-statistic: 0.3259 on 1 and 4896 DF, p-value: 0.5681Regression Model total.sulfur.dioxide and quality.
RegModel.total.sulfur.dioxide <- lm(total.sulfur.dioxide~quality, data=White_wines)
summary(RegModel.total.sulfur.dioxide)##
## Call:
## lm(formula = total.sulfur.dioxide ~ quality, data = White_wines)
##
## Residuals:
## Min 1Q Median 3Q Max
## -144.107 -28.722 -2.337 28.278 277.508
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 187.6464 4.0138 46.75 <2e-16 ***
## quality -8.3849 0.6752 -12.42 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 41.85 on 4896 degrees of freedom
## Multiple R-squared: 0.03053, Adjusted R-squared: 0.03034
## F-statistic: 154.2 on 1 and 4896 DF, p-value: < 2.2e-16Regression Model density and quality.
RegModel.density <- lm(density~quality, data=White_wines)
summary(RegModel.density)##
## Call:
## lm(formula = density ~ quality, data = White_wines)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.007718 -0.002104 -0.000361 0.001859 0.045079
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.000e+00 2.730e-04 3663.07 <2e-16 ***
## quality -1.037e-03 4.593e-05 -22.58 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.002847 on 4896 degrees of freedom
## Multiple R-squared: 0.09432, Adjusted R-squared: 0.09414
## F-statistic: 509.9 on 1 and 4896 DF, p-value: < 2.2e-16Regression Model pH and quality.
RegModel.pH <- lm(pH~quality, data=White_wines)
summary(RegModel.pH)##
## Call:
## lm(formula = pH ~ quality, data = White_wines)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.47034 -0.10034 -0.01034 0.08966 0.61966
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.088623 0.014413 214.301 < 2e-16 ***
## quality 0.016952 0.002425 6.992 3.08e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1503 on 4896 degrees of freedom
## Multiple R-squared: 0.009886, Adjusted R-squared: 0.009684
## F-statistic: 48.88 on 1 and 4896 DF, p-value: 3.081e-12Regression Model sulphates and quality.
RegModel.sulphates <- lm(sulphates~quality, data=White_wines)
summary(RegModel.sulphates)##
## Call:
## lm(formula = sulphates ~ quality, data = White_wines)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.27761 -0.08069 -0.01377 0.05931 0.58239
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.449189 0.010931 41.092 < 2e-16 ***
## quality 0.006917 0.001839 3.761 0.000171 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.114 on 4896 degrees of freedom
## Multiple R-squared: 0.002881, Adjusted R-squared: 0.002678
## F-statistic: 14.15 on 1 and 4896 DF, p-value: 0.000171From this we see neither citric acid nor free sulfur dioxide appear to have a significant linear relationship with quality.
Multiple Regressions
Now we will begin building a model using multiple regressions. However, prior to building the model, we will first split our dataset into a training and testing set.
set.seed(20170214) #Random Number seed is the date
White_wines$group <- runif(length(White_wines$quality), min = 0, max = 1) #create a new variable to add to dataset to distribute random numbers from 0-1
White_wines.train <- subset(White_wines, group <= 0.90) #assign 90% of the data to the training set
White_wines.test <- subset(White_wines, group > 0.90) #assign remaining data to the test set
#Did it work?
summary(White_wines.train)## fixed.acidity volatile.acidity citric.acid residual.sugar
## Min. : 3.800 Min. :0.0800 Min. :0.0000 Min. : 0.600
## 1st Qu.: 6.300 1st Qu.:0.2100 1st Qu.:0.2700 1st Qu.: 1.700
## Median : 6.800 Median :0.2600 Median :0.3200 Median : 5.100
## Mean : 6.851 Mean :0.2784 Mean :0.3337 Mean : 6.342
## 3rd Qu.: 7.300 3rd Qu.:0.3200 3rd Qu.:0.3900 3rd Qu.: 9.800
## Max. :14.200 Max. :1.1000 Max. :1.6600 Max. :65.800
## chlorides free.sulfur.dioxide total.sulfur.dioxide
## Min. :0.00900 Min. : 3.00 Min. : 9.0
## 1st Qu.:0.03600 1st Qu.: 23.00 1st Qu.:108.0
## Median :0.04300 Median : 34.00 Median :134.0
## Mean :0.04574 Mean : 35.28 Mean :138.3
## 3rd Qu.:0.05000 3rd Qu.: 46.00 3rd Qu.:167.0
## Max. :0.34600 Max. :289.00 Max. :440.0
## density pH sulphates alcohol
## Min. :0.9871 Min. :2.72 Min. :0.2200 Min. : 8.00
## 1st Qu.:0.9917 1st Qu.:3.09 1st Qu.:0.4100 1st Qu.: 9.50
## Median :0.9937 Median :3.18 Median :0.4700 Median :10.40
## Mean :0.9940 Mean :3.19 Mean :0.4892 Mean :10.52
## 3rd Qu.:0.9960 3rd Qu.:3.28 3rd Qu.:0.5500 3rd Qu.:11.40
## Max. :1.0390 Max. :3.82 Max. :1.0800 Max. :14.20
## quality group
## Min. :3.000 Min. :0.0002833
## 1st Qu.:5.000 1st Qu.:0.2285282
## Median :6.000 Median :0.4596618
## Mean :5.879 Mean :0.4570277
## 3rd Qu.:6.000 3rd Qu.:0.6859608
## Max. :9.000 Max. :0.8998507summary(White_wines.test)## fixed.acidity volatile.acidity citric.acid residual.sugar
## Min. : 5.000 Min. :0.0800 Min. :0.0000 Min. : 0.800
## 1st Qu.: 6.400 1st Qu.:0.2175 1st Qu.:0.2600 1st Qu.: 2.100
## Median : 6.800 Median :0.2600 Median :0.3200 Median : 6.300
## Mean : 6.889 Mean :0.2766 Mean :0.3387 Mean : 6.866
## 3rd Qu.: 7.300 3rd Qu.:0.3200 3rd Qu.:0.3900 3rd Qu.:10.400
## Max. :10.200 Max. :1.0050 Max. :0.8800 Max. :22.000
## chlorides free.sulfur.dioxide total.sulfur.dioxide
## Min. :0.01400 Min. : 2.0 Min. : 24.0
## 1st Qu.:0.03675 1st Qu.: 23.0 1st Qu.:108.0
## Median :0.04300 Median : 35.0 Median :135.0
## Mean :0.04612 Mean : 35.6 Mean :139.4
## 3rd Qu.:0.05000 3rd Qu.: 47.0 3rd Qu.:170.2
## Max. :0.20400 Max. :124.0 Max. :260.0
## density pH sulphates alcohol
## Min. :0.9877 Min. :2.770 Min. :0.280 Min. : 8.40
## 1st Qu.:0.9918 1st Qu.:3.080 1st Qu.:0.400 1st Qu.: 9.40
## Median :0.9941 Median :3.170 Median :0.480 Median :10.20
## Mean :0.9942 Mean :3.174 Mean :0.496 Mean :10.45
## 3rd Qu.:0.9964 3rd Qu.:3.260 3rd Qu.:0.560 3rd Qu.:11.30
## Max. :1.0010 Max. :3.690 Max. :1.010 Max. :13.90
## quality group
## Min. :3.000 Min. :0.9001
## 1st Qu.:5.000 1st Qu.:0.9229
## Median :6.000 Median :0.9528
## Mean :5.872 Mean :0.9506
## 3rd Qu.:6.000 3rd Qu.:0.9758
## Max. :8.000 Max. :0.9993Now we will begin with a full model including all variables.
LinearModel.Full <- lm(quality ~ alcohol + chlorides + citric.acid +
density + fixed.acidity + free.sulfur.dioxide + pH + residual.sugar
+ sulphates + total.sulfur.dioxide + volatile.acidity,
data=White_wines.train)
summary(LinearModel.Full)##
## Call:
## lm(formula = quality ~ alcohol + chlorides + citric.acid + density +
## fixed.acidity + free.sulfur.dioxide + pH + residual.sugar +
## sulphates + total.sulfur.dioxide + volatile.acidity, data = White_wines.train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.8642 -0.4973 -0.0362 0.4704 3.0782
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.552e+02 1.937e+01 8.013 1.42e-15 ***
## alcohol 1.885e-01 2.510e-02 7.510 7.10e-14 ***
## chlorides -2.444e-01 5.701e-01 -0.429 0.668114
## citric.acid 4.294e-02 1.010e-01 0.425 0.670887
## density -1.555e+02 1.965e+01 -7.917 3.06e-15 ***
## fixed.acidity 8.103e-02 2.176e-02 3.724 0.000199 ***
## free.sulfur.dioxide 4.064e-03 8.870e-04 4.581 4.74e-06 ***
## pH 7.268e-01 1.099e-01 6.614 4.19e-11 ***
## residual.sugar 8.492e-02 7.816e-03 10.865 < 2e-16 ***
## sulphates 6.578e-01 1.068e-01 6.156 8.10e-10 ***
## total.sulfur.dioxide -4.434e-04 3.963e-04 -1.119 0.263311
## volatile.acidity -1.822e+00 1.199e-01 -15.199 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7538 on 4426 degrees of freedom
## Multiple R-squared: 0.2805, Adjusted R-squared: 0.2787
## F-statistic: 156.9 on 11 and 4426 DF, p-value: < 2.2e-16The full model can be used to explain 28% of the variability in taste. The F statistic is 156.9 and is highly significant. We will investigate what occurs as this model is reduced.
To continue we will use the backwards selection strategy and remove all variable that were not significant in the full model.
Reduced Model 1 will include alcohol, density, fixed acidity, free sulfur dioxide, pH, residal sugar, sulphates, volatile acidity.
LinearModel.2 <- lm(quality ~ alcohol + density + fixed.acidity +
free.sulfur.dioxide + pH + residual.sugar + sulphates + volatile.acidity,
data=White_wines.train)
summary(LinearModel.2)##
## Call:
## lm(formula = quality ~ alcohol + density + fixed.acidity + free.sulfur.dioxide +
## pH + residual.sugar + sulphates + volatile.acidity, data = White_wines.train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.8536 -0.4930 -0.0388 0.4675 3.0889
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.598e+02 1.872e+01 8.535 < 2e-16 ***
## alcohol 1.888e-01 2.495e-02 7.566 4.64e-14 ***
## density -1.603e+02 1.898e+01 -8.445 < 2e-16 ***
## fixed.acidity 8.386e-02 2.133e-02 3.931 8.58e-05 ***
## free.sulfur.dioxide 3.487e-03 7.137e-04 4.885 1.07e-06 ***
## pH 7.325e-01 1.078e-01 6.792 1.25e-11 ***
## residual.sugar 8.639e-02 7.594e-03 11.377 < 2e-16 ***
## sulphates 6.524e-01 1.064e-01 6.130 9.57e-10 ***
## volatile.acidity -1.861e+00 1.152e-01 -16.150 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7537 on 4429 degrees of freedom
## Multiple R-squared: 0.2802, Adjusted R-squared: 0.2789
## F-statistic: 215.5 on 8 and 4429 DF, p-value: < 2.2e-16This reduced model can still be used to explain 28% of the variability in taste. The F statistic increased to 215.5 and is highly significant. We will investigate what occurs as this model is reduced.
View influential variables
#added variable plots
avPlots(LinearModel.2, id.n=2, id.cex=0.7)
#id.n - identify n most influential observations so you can pick out outlier values labeling them as farmers babysitters etc
#id.cex - controls the size of the dotLook at residuals
# run the qq-plot
qqPlot(LinearModel.2, id.n=3)
## 4746 254 2782
## 1 2 4438# here, id.n identifies the n observations with the largest residuals in absolute value# diagnostics for the first model with 3 independent variables
residualPlots(LinearModel.2)
## Test stat Pr(>|t|)
## alcohol 5.191 0.000
## density 5.552 0.000
## fixed.acidity -4.163 0.000
## free.sulfur.dioxide -10.160 0.000
## pH 0.880 0.379
## residual.sugar 2.520 0.012
## sulphates 0.729 0.466
## volatile.acidity 3.184 0.001
## Tukey test 2.551 0.011Outlier
#run Bonferroni test for outliers
outlierTest(LinearModel.2)## rstudent unadjusted p-value Bonferonni p
## 4746 -5.285819 1.3116e-07 0.00058211
## 2782 4.931011 8.4800e-07 0.00376340
## 254 -4.496908 7.0712e-06 0.03138200
## 446 -4.485892 7.4449e-06 0.03304100Influence Plot
#make influence plot
influencePlot(LinearModel.2, id.n=3)
## StudRes Hat CookD
## 254 -4.4969082 0.002555952 0.005732828
## 1527 -0.6554449 0.038237083 0.001898028
## 1932 -3.7826585 0.015025299 0.024179469
## 2782 4.9310113 0.351726593 1.458130190
## 4746 -5.2858195 0.058668586 0.192314295Heteroskedascity
#test for heteroskedasticity
ncvTest(LinearModel.2) #tests for non constant variance. ## Non-constant Variance Score Test
## Variance formula: ~ fitted.values
## Chisquare = 16.04371 Df = 1 p = 6.189685e-05vif(LinearModel.2)## alcohol density fixed.acidity
## 7.337053 25.278419 2.545179
## free.sulfur.dioxide pH residual.sugar
## 1.151922 2.083100 11.620123
## sulphates volatile.acidity
## 1.126483 1.060061#if higher than 4 we want to take variable out b/c it is not independent and highly correlates with something in there Based on the previous plots/analysis we further reduce the model. will remove density, and residual sugar, and free sulfur dioxide from the analysis.
LinearModel.3 <- lm(quality ~ alcohol + fixed.acidity +
free.sulfur.dioxide + pH + residual.sugar + sulphates + volatile.acidity,
data=White_wines.train)
summary(LinearModel.3)##
## Call:
## lm(formula = quality ~ alcohol + fixed.acidity + free.sulfur.dioxide +
## pH + residual.sugar + sulphates + volatile.acidity, data = White_wines.train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.8938 -0.4962 -0.0333 0.4624 3.1774
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.7101754 0.3536637 4.836 1.37e-06 ***
## alcohol 0.3800365 0.0105722 35.947 < 2e-16 ***
## fixed.acidity -0.0451858 0.0150020 -3.012 0.002610 **
## free.sulfur.dioxide 0.0037010 0.0007189 5.148 2.74e-07 ***
## pH 0.1719520 0.0856708 2.007 0.044797 *
## residual.sugar 0.0261723 0.0026316 9.946 < 2e-16 ***
## sulphates 0.3989868 0.1029242 3.877 0.000107 ***
## volatile.acidity -2.0134448 0.1146832 -17.557 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7597 on 4430 degrees of freedom
## Multiple R-squared: 0.2686, Adjusted R-squared: 0.2675
## F-statistic: 232.5 on 7 and 4430 DF, p-value: < 2.2e-16This reduced model can still be used to explain 27% of the variability in taste. The F statistic increased to 232.5 and is still highly significant.
# diagnostics for the first model with 3 independent variables
residualPlots(LinearModel.3)
## Test stat Pr(>|t|)
## alcohol 5.243 0.000
## fixed.acidity -3.584 0.000
## free.sulfur.dioxide -10.370 0.000
## pH 0.386 0.700
## residual.sugar -2.049 0.041
## sulphates 0.878 0.380
## volatile.acidity 1.968 0.049
## Tukey test 0.145 0.884We will investigate what occurs as this model is further reduced by removing free sulfur dioxide.
LinearModel.4 <- lm(quality ~ alcohol + fixed.acidity + pH + residual.sugar + sulphates + volatile.acidity,
data=White_wines.train)
summary(LinearModel.4)##
## Call:
## lm(formula = quality ~ alcohol + fixed.acidity + pH + residual.sugar +
## sulphates + volatile.acidity, data = White_wines.train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4043 -0.4962 -0.0369 0.4662 3.1503
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.914453 0.352441 5.432 5.87e-08 ***
## alcohol 0.373244 0.010520 35.481 < 2e-16 ***
## fixed.acidity -0.051461 0.014995 -3.432 0.000605 ***
## pH 0.178963 0.085906 2.083 0.037286 *
## residual.sugar 0.029424 0.002562 11.485 < 2e-16 ***
## sulphates 0.432488 0.103013 4.198 2.74e-05 ***
## volatile.acidity -2.080385 0.114271 -18.206 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7618 on 4431 degrees of freedom
## Multiple R-squared: 0.2643, Adjusted R-squared: 0.2633
## F-statistic: 265.3 on 6 and 4431 DF, p-value: < 2.2e-16This reduced model can still be used to explain 26% of the variability in taste. The F statistic increased to 265.3 and is still highly significant.
# diagnostics for the first model with 3 independent variables
residualPlots(LinearModel.4)
## Test stat Pr(>|t|)
## alcohol 5.496 0.000
## fixed.acidity -3.795 0.000
## pH 0.165 0.869
## residual.sugar -2.603 0.009
## sulphates 1.054 0.292
## volatile.acidity 1.760 0.078
## Tukey test -0.203 0.839vif(LinearModel.4)## alcohol fixed.acidity pH residual.sugar
## 1.276125 1.230980 1.293699 1.294518
## sulphates volatile.acidity
## 1.032790 1.020648#if higher than 4 we want to take variable out b/c it is not independent and highly correlates with something in there Residual sugar
LinearModel.5 <- lm(quality ~ alcohol + fixed.acidity + residual.sugar + sulphates + volatile.acidity,
data=White_wines.train)
summary(LinearModel.5)##
## Call:
## lm(formula = quality ~ alcohol + fixed.acidity + residual.sugar +
## sulphates + volatile.acidity, data = White_wines.train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.3580 -0.4939 -0.0352 0.4642 3.1857
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.560289 0.167709 15.27 < 2e-16 ***
## alcohol 0.373620 0.010522 35.51 < 2e-16 ***
## fixed.acidity -0.064489 0.013634 -4.73 2.31e-06 ***
## residual.sugar 0.028655 0.002536 11.30 < 2e-16 ***
## sulphates 0.468359 0.101603 4.61 4.15e-06 ***
## volatile.acidity -2.088819 0.114243 -18.28 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7621 on 4432 degrees of freedom
## Multiple R-squared: 0.2635, Adjusted R-squared: 0.2627
## F-statistic: 317.2 on 5 and 4432 DF, p-value: < 2.2e-16This reduced model can still be used to explain 26% of the variability in taste. The F statistic increased to 317.2 and is still highly significant. An increase in the adjusted R squared indicates it may fit better than the previous model.
# diagnostics for the first model with 3 independent variables
residualPlots(LinearModel.5)
## Test stat Pr(>|t|)
## alcohol 5.143 0.000
## fixed.acidity -3.568 0.000
## residual.sugar -2.491 0.013
## sulphates 0.929 0.353
## volatile.acidity 1.930 0.054
## Tukey test -0.471 0.637vif(LinearModel.5)## alcohol fixed.acidity residual.sugar sulphates
## 1.275751 1.016852 1.267666 1.003936
## volatile.acidity
## 1.019367#if higher than 4 we want to take variable out b/c it is not independent and highly correlates with something in there Based on the residuals I would like to see what happens when residual sugar and fixed acidity are removed from the model.
LinearModel.5 <- lm(quality ~ alcohol + sulphates + volatile.acidity,
data=White_wines.train)
summary(LinearModel.5)##
## Call:
## lm(formula = quality ~ alcohol + sulphates + volatile.acidity,
## data = White_wines.train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.3158 -0.4886 -0.0468 0.4947 3.1571
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.785383 0.116676 23.873 < 2e-16 ***
## alcohol 0.325036 0.009493 34.239 < 2e-16 ***
## sulphates 0.434956 0.103148 4.217 2.53e-05 ***
## volatile.acidity -1.936946 0.115352 -16.792 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7744 on 4434 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2388
## F-statistic: 465 on 3 and 4434 DF, p-value: < 2.2e-16This model does not appear to fit better than the previous model. It only accounts for 24% of the variability and while still significant the adjust R squared value has decreased from .26 to .23.
compareCoefs(LinearModel.2, LinearModel.3, LinearModel.4, LinearModel.5,
LinearModel.Full)##
## Call:
## 1: lm(formula = quality ~ alcohol + density + fixed.acidity +
## free.sulfur.dioxide + pH + residual.sugar + sulphates +
## volatile.acidity, data = White_wines.train)
## 2: lm(formula = quality ~ alcohol + fixed.acidity +
## free.sulfur.dioxide + pH + residual.sugar + sulphates +
## volatile.acidity, data = White_wines.train)
## 3: lm(formula = quality ~ alcohol + fixed.acidity + pH +
## residual.sugar + sulphates + volatile.acidity, data =
## White_wines.train)
## 4: lm(formula = quality ~ alcohol + sulphates + volatile.acidity, data
## = White_wines.train)
## 5: lm(formula = quality ~ alcohol + chlorides + citric.acid + density
## + fixed.acidity + free.sulfur.dioxide + pH + residual.sugar +
## sulphates + total.sulfur.dioxide + volatile.acidity, data =
## White_wines.train)
## Est. 1 SE 1 Est. 2 SE 2 Est. 3
## (Intercept) 1.60e+02 1.87e+01 1.71e+00 3.54e-01 1.91e+00
## alcohol 1.89e-01 2.50e-02 3.80e-01 1.06e-02 3.73e-01
## density -1.60e+02 1.90e+01
## fixed.acidity 8.39e-02 2.13e-02 -4.52e-02 1.50e-02 -5.15e-02
## free.sulfur.dioxide 3.49e-03 7.14e-04 3.70e-03 7.19e-04
## pH 7.32e-01 1.08e-01 1.72e-01 8.57e-02 1.79e-01
## residual.sugar 8.64e-02 7.59e-03 2.62e-02 2.63e-03 2.94e-02
## sulphates 6.52e-01 1.06e-01 3.99e-01 1.03e-01 4.32e-01
## volatile.acidity -1.86e+00 1.15e-01 -2.01e+00 1.15e-01 -2.08e+00
## chlorides
## citric.acid
## total.sulfur.dioxide
## SE 3 Est. 4 SE 4 Est. 5 SE 5
## (Intercept) 3.52e-01 2.79e+00 1.17e-01 1.55e+02 1.94e+01
## alcohol 1.05e-02 3.25e-01 9.49e-03 1.88e-01 2.51e-02
## density -1.56e+02 1.96e+01
## fixed.acidity 1.50e-02 8.10e-02 2.18e-02
## free.sulfur.dioxide 4.06e-03 8.87e-04
## pH 8.59e-02 7.27e-01 1.10e-01
## residual.sugar 2.56e-03 8.49e-02 7.82e-03
## sulphates 1.03e-01 4.35e-01 1.03e-01 6.58e-01 1.07e-01
## volatile.acidity 1.14e-01 -1.94e+00 1.15e-01 -1.82e+00 1.20e-01
## chlorides -2.44e-01 5.70e-01
## citric.acid 4.29e-02 1.01e-01
## total.sulfur.dioxide -4.43e-04 3.96e-04# compare the results of the two regression models
stargazer(LinearModel.3,LinearModel.4, LinearModel.5,title="Comparison of Regression outputs",type="text",align=TRUE)Comparison of Regression outputs
Dependent variable:
-----------------------------------------------------------------------------
quality
(1) (2) (3) | alcohol 0.380*** 0.373*** 0.325*** (0.011) (0.011) (0.009) |
| fixed.acidity -0.045*** -0.051*** (0.015) (0.015) |
| free.sulfur.dioxide 0.004*** (0.001) |
| pH 0.172** 0.179** (0.086) (0.086) |
| residual.sugar 0.026*** 0.029*** (0.003) (0.003) |
| sulphates 0.399*** 0.432*** 0.435*** (0.103) (0.103) (0.103) |
| volatile.acidity -2.013*** -2.080*** -1.937*** (0.115) (0.114) (0.115) |
| Constant 1.710*** 1.914*** 2.785*** (0.354) (0.352) (0.117) |
Observations 4,438 4,438 4,438
R2 0.269 0.264 0.239
Adjusted R2 0.267 0.263 0.239
Residual Std. Error 0.760 (df = 4430) 0.762 (df = 4431) 0.774 (df = 4434)
F Statistic 232.461*** (df = 7; 4430) 265.260*** (df = 6; 4431) 465.047*** (df = 3; 4434) ================================================================================================= Note: p<0.1; p<0.05; p<0.01
#can only be seen when knitting to html if you change type to text you can see the table now type=html or text or latek as options#test for heteroskedasticity
ncvTest(LinearModel.4) #tests for non constant variance. All biomarkers fail this test. since p is big its a homoskedastic set## Non-constant Variance Score Test
## Variance formula: ~ fitted.values
## Chisquare = 26.78513 Df = 1 p = 2.273799e-07vif(LinearModel.4)## alcohol fixed.acidity pH residual.sugar
## 1.276125 1.230980 1.293699 1.294518
## sulphates volatile.acidity
## 1.032790 1.020648#if higher than 4 we want to take variable out b/c it is not independent and highly correlates with something in there #make influence plot
influencePlot(LinearModel.4, id.n=3)
## StudRes Hat CookD
## 254 -4.4819619 0.001694585 0.0048503308
## 446 -4.2795152 0.001112097 0.0029015056
## 741 -4.3948589 0.001592595 0.0043832688
## 1418 -3.5659407 0.004188520 0.0076205600
## 1527 0.5376494 0.020892764 0.0008813241
## 2051 -3.3491187 0.008253723 0.0133049158
## 2782 -0.9626167 0.053125121 0.0074271681
## 4040 -1.1545176 0.015677406 0.0030325443
## 4481 -3.0340351 0.006500608 0.0085886727Based on this data I believe LinearModel.4 to be the best model of this data. I am not pleased with the plots of the residuals or influential points. I would also like to include less variables in the model. However I am not sure how much of to trade for the amount of variability accounted for by the model and these other things. Currently the model accounts for 26% of the variability in the score for quality.
We will now run the model on the testing dataset to see if it is a good predictor of the values.
predict(LinearModel.4, White_wines.test)## 5 22 30 49 73 94 96 97
## 5.754555 5.849885 6.389817 5.543937 5.885102 6.637284 5.532576 5.428932
## 108 128 144 146 153 156 160 161
## 5.773299 5.775722 5.585470 5.866041 5.367857 5.467006 6.032974 6.377618
## 170 178 192 202 208 216 226 233
## 5.197689 5.776077 5.617164 5.320064 5.107399 5.610988 5.583584 5.519034
## 259 261 269 286 295 318 322 323
## 6.374263 6.061912 5.098319 5.123943 4.180785 5.952060 5.952060 5.877922
## 338 351 355 360 379 388 390 405
## 5.656004 6.562630 5.324585 5.982613 5.617402 5.566075 5.672618 6.860365
## 422 428 429 442 447 451 465 466
## 6.070248 5.608655 5.254582 5.774928 6.551588 4.759324 5.085239 5.496366
## 498 503 534 550 562 572 575 584
## 6.620995 5.563136 5.507412 4.887359 5.445372 6.178006 6.298038 5.637185
## 586 591 616 624 631 634 648 657
## 5.637185 5.443775 5.357665 6.130614 5.649293 5.696528 5.758915 5.968483
## 661 663 676 702 706 708 714 726
## 5.259055 4.858217 5.361469 7.109821 5.822284 6.307564 5.673433 5.378833
## 728 748 781 837 844 854 855 857
## 5.651501 6.138997 5.772917 6.276716 6.053175 5.106925 5.765050 5.398917
## 867 876 878 879 883 891 895 902
## 5.765050 5.728577 5.612588 5.363674 6.259618 5.758706 6.001125 5.649969
## 905 912 917 922 945 980 988 992
## 5.221398 5.604962 6.089411 5.236697 5.465398 5.656118 5.659932 5.627005
## 1000 1002 1011 1012 1028 1055 1076 1090
## 5.697409 5.463115 5.957041 6.280983 4.828393 5.576456 6.397113 6.014210
## 1091 1106 1109 1137 1146 1147 1156 1167
## 5.784013 6.242261 6.545170 6.590166 5.180613 5.592791 5.637328 4.950176
## 1173 1182 1213 1214 1223 1233 1235 1241
## 6.761323 5.406320 5.925312 5.821558 5.563886 6.266618 5.613622 5.506863
## 1261 1266 1299 1300 1312 1315 1316 1341
## 5.444045 6.162557 6.406539 6.169369 6.010110 5.356462 5.853127 5.694987
## 1350 1366 1370 1394 1395 1397 1404 1405
## 5.271930 5.842960 4.820627 5.720809 6.854030 6.197566 6.015806 6.013556
## 1407 1433 1442 1445 1454 1456 1473 1481
## 5.856961 6.199309 5.576169 5.931977 5.984055 5.502641 6.055716 6.032355
## 1518 1523 1541 1586 1624 1632 1633 1659
## 5.906402 6.123471 5.887483 5.272583 5.702082 6.193954 6.572516 5.193990
## 1670 1677 1704 1710 1717 1757 1758 1762
## 5.386829 6.062452 5.410414 5.949095 6.124591 5.476755 5.251264 5.639551
## 1765 1768 1790 1804 1811 1815 1871 1880
## 5.476755 5.125231 6.531564 5.805759 5.311551 6.192163 5.900968 5.741693
## 1882 1883 1891 1926 1928 1936 1938 1942
## 5.741693 5.590267 5.590267 5.699303 5.718286 5.570971 6.293422 5.228709
## 1952 1977 1985 1988 1993 2014 2020 2029
## 3.914879 5.871837 5.607854 5.795175 5.889762 6.148233 6.148233 5.598355
## 2044 2073 2088 2091 2107 2121 2129 2133
## 6.105045 5.720818 5.495504 5.448895 5.610094 5.428639 4.911079 5.939254
## 2141 2142 2149 2163 2183 2190 2200 2201
## 5.770479 5.770479 5.399303 4.650740 5.745360 5.798194 5.737759 5.507977
## 2210 2216 2271 2290 2309 2312 2329 2383
## 6.330326 5.728844 6.020957 6.417134 5.134960 6.432918 5.485319 6.037306
## 2385 2386 2392 2396 2404 2417 2423 2434
## 6.037306 5.457223 5.954604 5.840390 6.530634 6.048425 5.288937 5.309192
## 2456 2477 2481 2498 2525 2552 2558 2560
## 5.536748 5.124726 5.578971 5.737561 5.512502 5.605980 6.366352 5.388761
## 2563 2584 2586 2589 2621 2638 2643 2650
## 5.833609 5.687656 5.687656 5.687656 5.571208 5.401601 5.731640 6.036431
## 2657 2668 2674 2682 2691 2703 2715 2746
## 5.870158 6.313997 5.990224 5.739648 5.288876 5.626365 5.581857 6.370984
## 2753 2766 2776 2784 2785 2788 2793 2795
## 6.319638 6.250635 6.163672 6.080017 6.080017 5.860250 5.566642 5.364473
## 2799 2801 2809 2822 2837 2887 2891 2924
## 5.982030 5.512134 5.469431 5.506068 5.539831 5.425472 6.551646 6.081222
## 2932 2943 2945 2957 2959 2960 2961 2962
## 6.766951 5.509591 5.916456 6.492353 5.739704 6.535618 6.613038 6.231767
## 3015 3043 3052 3058 3062 3066 3084 3105
## 6.020093 5.748245 5.386282 6.804767 6.036641 6.100894 7.023438 6.074370
## 3108 3115 3117 3127 3128 3136 3138 3139
## 5.823993 6.343635 6.109906 6.249132 5.915144 5.239467 6.007609 6.087446
## 3161 3166 3169 3180 3181 3193 3209 3233
## 6.514180 5.480466 6.717271 5.700652 6.436530 6.512904 6.005123 6.725933
## 3238 3252 3267 3292 3294 3298 3300 3308
## 6.089750 5.878405 6.449780 6.982516 6.510050 5.479262 6.429623 5.970557
## 3323 3330 3335 3344 3349 3368 3369 3381
## 6.646428 6.036378 5.737937 5.551329 6.233818 6.707571 5.812229 6.414225
## 3399 3407 3413 3426 3427 3433 3437 3459
## 5.577143 6.114846 5.891664 5.955132 5.955132 6.025953 6.446375 7.153511
## 3468 3487 3502 3509 3511 3516 3525 3526
## 5.401801 6.097220 6.208526 5.592099 6.114772 6.617752 6.183876 6.183876
## 3528 3541 3542 3549 3567 3569 3570 3571
## 5.650672 6.576099 5.971824 6.223617 6.273472 6.311837 6.139212 6.161398
## 3591 3600 3606 3611 3612 3616 3628 3632
## 6.565801 5.394465 5.356274 5.356274 6.313549 6.430932 5.291766 5.929097
## 3633 3644 3648 3667 3669 3675 3696 3702
## 5.721435 6.219669 6.583262 6.173623 6.227363 5.125455 6.278390 5.765670
## 3707 3710 3712 3721 3729 3733 3736 3748
## 6.405111 5.484367 5.586235 6.257105 7.155641 5.610276 6.910907 5.476774
## 3771 3775 3789 3797 3801 3820 3821 3824
## 6.344864 5.515612 5.573958 5.957831 5.815028 6.194149 5.769951 5.591405
## 3828 3840 3844 3850 3861 3864 3871 3872
## 6.240179 4.955945 6.654612 5.864755 5.382777 5.694787 5.956016 5.694787
## 3873 3931 3948 3954 3977 3981 4023 4041
## 5.150634 6.298004 6.564024 5.887486 6.734503 6.240065 5.193307 5.437248
## 4045 4055 4060 4066 4074 4078 4116 4134
## 5.609037 5.777976 6.278882 6.604508 4.781410 6.096600 6.256166 6.200459
## 4138 4151 4166 4168 4180 4193 4251 4253
## 5.277407 5.123081 6.412160 6.920872 5.892801 5.853021 6.158141 6.158141
## 4270 4276 4280 4289 4295 4314 4328 4345
## 5.289500 5.731301 6.251883 5.944465 6.164465 6.238063 5.647516 5.192819
## 4351 4355 4359 4360 4387 4398 4400 4402
## 6.659673 6.080761 5.657107 5.422664 6.108683 5.793002 5.790480 6.630611
## 4403 4420 4422 4425 4426 4430 4437 4451
## 6.158496 5.660858 5.660858 5.660858 5.820353 5.857527 6.413331 6.023727
## 4456 4476 4480 4488 4503 4516 4523 4528
## 5.728444 6.681839 4.758790 6.367368 6.134158 6.032092 5.513647 5.719944
## 4533 4552 4562 4568 4583 4629 4640 4646
## 6.396449 6.031559 6.428608 6.225140 5.916541 6.135481 5.545675 6.812276
## 4657 4666 4667 4675 4676 4678 4684 4690
## 6.221562 6.793892 5.353244 6.174447 6.129973 6.417458 6.184083 5.634909
## 4697 4701 4715 4716 4736 4745 4750 4764
## 7.162050 5.771801 6.273579 5.945358 6.142887 5.940731 5.708126 6.153141
## 4766 4771 4783 4817 4824 4830 4832 4834
## 6.763756 5.750237 5.396718 5.781465 6.383357 6.488271 5.249450 5.918841
## 4837 4858 4867 4885
## 5.219486 5.996485 5.831067 5.498981LinearModel.test <- lm(quality ~ alcohol + fixed.acidity + pH + residual.sugar + sulphates + volatile.acidity,
data=White_wines.test)
summary(LinearModel.test)##
## Call:
## lm(formula = quality ~ alcohol + fixed.acidity + pH + residual.sugar +
## sulphates + volatile.acidity, data = White_wines.test)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.72234 -0.48958 -0.04004 0.47059 2.60770
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.095325 1.116756 3.667 0.000274 ***
## alcohol 0.339922 0.031396 10.827 < 2e-16 ***
## fixed.acidity -0.171231 0.046668 -3.669 0.000272 ***
## pH -0.068681 0.265655 -0.259 0.796111
## residual.sugar 0.014961 0.007941 1.884 0.060183 .
## sulphates 0.257427 0.271472 0.948 0.343502
## volatile.acidity -2.201534 0.347204 -6.341 5.54e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7241 on 453 degrees of freedom
## Multiple R-squared: 0.3124, Adjusted R-squared: 0.3033
## F-statistic: 34.3 on 6 and 453 DF, p-value: < 2.2e-16Running this model on the test data provides a significant F statistic, with the model explaining 30% of the variability in the score for quality. This model uses alcohol, fixed acidity, pH, residual sugar, sulphates, volatile acidity to explain quality. This equation for this model is:
Y = 4.10 + (0.34)x1 + (-0.17)x2 + (-0.07)x3 + (0.01)x4 + (0.25)x5 + (-2.20)x6 + E
Where: Y= quality x1= alcohol x2= fixed acidity x3= pH x4= residual sugar x5= sulphates x6= volatile acid E= Error
Using this model it appears volatile acid influences quality the most. When keeping the other variables constant a 1 point change in volatile acid will cause a -2.20 change in the quality score of the wine.